He was a leading figure at the MIT Radiation Laboratory, and later he was head at the Columbia University laboratory that carried out the crucial Lamb-Retherford experiment and the anomalous magnetic dipole moment of the electron published at The Magnetic Moment of the Electron by Kusch and Foley (1948) using related techniques.

Basically the operators are just analogous to the classical ones e.g. the classical:becomes:

Besides the angular momentum in each direction, we also have the total angular momentum:

Then you have to understand what each one of those does to the each atomic orbital:

There is an uncertainty principle between the x, y and z angular momentums, we can only measure one of them with certainty at a time. Video 1. "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)" justifies this intuitively by mentioning that this is analogous to precession: if you try to measure electrons e.g. with the Zeeman effect the precess on the other directions which you end up modifing.

TODO experiment. Likely Zeeman effect.

Feynman became a terrible womanizer after his first wife Arline Greenbaum died, involving himself with several married women, and leading to at least two abortions according to Genius: Richard Feynman and Modern Physics by James Gleick (1994).

Ciro Santilli likes to think that he is quite liberal and not a strict follower of Christian morals, but this one shocked him slightly even. Feynman could be a God, but he could also be a dick sometimes.

One particular case that stuck to Ciro Santilli's mind, partly because he is Brazilian, is when Feynman was in Brazil, he had a girlfriend called Clotilde that called him "Ricardinho", which means "Little Richard"; -inho is a diminutive suffix in Portuguese, and also indicates affection. At some point he even promised to take her back to the United States, but didn't in the end, and instead came back and married his second wife in marriage that soon failed.

Richard's third and final wife, Gweneth Howarth, seemed a good match for him though. When they started courting, she made it very clear that Feynman should decide if he wanted her or not soon, because she had other options available and being actively tested. Fight fire with fire.

2023 full connectome published on Science by Marta Zlatic's lab at the MRC:

The hard part then is how to make any predictions from it:

2024: www.nature.com/articles/d41586-024-03190-y Largest brain map ever reveals fruit fly's neurons in exquisite detail

As of 2022, it had been almost fully decoded by post mortem connectome extraction with microtome!!! 135k neurons.

That article mentions the humongous paper elifesciences.org/articles/66039 elifesciences.org/articles/66039 "A connectome of the Drosophila central complex reveals network motifs suitable for flexible navigation and context-dependent action selection" by a group from Janelia Research Campus. THe paper is so large that it makes eLife hang.

Mathematically, we can decide if two events are timelike-separated or spacelike-separated by just looking at the sign of the spacetime interval between them.

On the light cone, these are events on the left/right part of the cone.

Different observers might not agree on the order of two spacelike-separated events.

Further discussion at Section "Light cone".

The opposite of those events are timelike-separated events.

For this sub-case, we can define the Lie algebra of a Lie group $G$ as the set of all matrices $M\in G$ such that for all $t\in \mathbb{R}$:If we fix a given $M$ and vary $t$, we obtain a subgroup of $G$. This type of subgroup is known as a one parameter subgroup.

The immediate question is then if every element of $G$ can be reached in a unique way (i.e. is the exponential map a bijection). By looking at the matrix logarithm however we conclude that this is not the case for real matrices, but it is for complex matrices.

Examples:

TODO example it can be seen that the Lie algebra is not closed matrix multiplication, even though the corresponding group is by definition. But it is closed under the Lie bracket operation.

Discrete quantum system model that can model both spin in the Stern-Gerlach experiment or photon polarization in polarizer.

Also known in quantum computing as a qubit :-)

Before reading any further, you must understand heat equation solution with Fourier series, which uses separation of variables.

Once that example is clear, we see that the exact same separation of variables can be done to the Schrödinger equation. If we name the constant of the separation of variables $E$ for energy, we get:

Because the time part of the equation is always the same and always trivial to solve, all we have to do to actually solve the Schrodinger equation is to solve the time independent one, and then we can construct the full solution trivially.

Once we've solved the time-independent part for each possible $E$, we can construct a solution exactly as we did in heat equation solution with Fourier series: we make a weighted sum over all possible $E$ to match the initial condition, which is analogous to the Fourier series in the case of the heat equation to reach a final full solution:

The fact that this approximation of the initial condition is always possible from is mathematically proven by some version of the spectral theorem based on the fact that The Schrodinger equation Hamiltonian has to be Hermitian and therefore behaves nicely.

It is interesting to note that solving the time-independent Schrodinger equation can also be seen exactly as an eigenvalue equation where:The only difference from usual matrix eigenvectors is that we are now dealing with an infinite dimensional vector space.

Furthermore:

Bibliography:

TODO find/create decent answer.

I think the best answer is something along:

A basic non-precise intuition is that a good model of reality is that electrons do not "interact with one another directly via the electromagnetic field".

A better model happens to be the quantum field theory view that the electromagnetic field interacts with the photon field but not directly with itself, and then the photon field interacts with parts of the electromagnetic field further away.

The more precise statement is that the photon field is a gauge field of the electromagnetic force under local U(1) symmetry, which is described by a Lie group. TODO understand.

This idea was first applied in general relativity, where Einstein understood that the "force of gravity" can be understood just in terms of symmetry and curvature of space. This was later applied o quantum electrodynamics and the entire Standard Model.

From Video "Lorenzo Sadun on the "Yang-Mills and Mass Gap" Millennium problem":

Bibliography:

TODO clickbait, or is it that good?

Like the rest of the Standard Model Lagrangian, this can be split into two parts: